--- a/src/HOL/SetInterval.thy Tue Apr 03 14:09:37 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1439 +0,0 @@
-(* Title: HOL/SetInterval.thy
- Author: Tobias Nipkow
- Author: Clemens Ballarin
- Author: Jeremy Avigad
-
-lessThan, greaterThan, atLeast, atMost and two-sided intervals
-*)
-
-header {* Set intervals *}
-
-theory SetInterval
-imports Int Nat_Transfer
-begin
-
-context ord
-begin
-
-definition
- lessThan :: "'a => 'a set" ("(1{..<_})") where
- "{..<u} == {x. x < u}"
-
-definition
- atMost :: "'a => 'a set" ("(1{.._})") where
- "{..u} == {x. x \<le> u}"
-
-definition
- greaterThan :: "'a => 'a set" ("(1{_<..})") where
- "{l<..} == {x. l<x}"
-
-definition
- atLeast :: "'a => 'a set" ("(1{_..})") where
- "{l..} == {x. l\<le>x}"
-
-definition
- greaterThanLessThan :: "'a => 'a => 'a set" ("(1{_<..<_})") where
- "{l<..<u} == {l<..} Int {..<u}"
-
-definition
- atLeastLessThan :: "'a => 'a => 'a set" ("(1{_..<_})") where
- "{l..<u} == {l..} Int {..<u}"
-
-definition
- greaterThanAtMost :: "'a => 'a => 'a set" ("(1{_<.._})") where
- "{l<..u} == {l<..} Int {..u}"
-
-definition
- atLeastAtMost :: "'a => 'a => 'a set" ("(1{_.._})") where
- "{l..u} == {l..} Int {..u}"
-
-end
-
-
-text{* A note of warning when using @{term"{..<n}"} on type @{typ
-nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
-@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
-
-syntax
- "_UNION_le" :: "'a => 'a => 'b set => 'b set" ("(3UN _<=_./ _)" [0, 0, 10] 10)
- "_UNION_less" :: "'a => 'a => 'b set => 'b set" ("(3UN _<_./ _)" [0, 0, 10] 10)
- "_INTER_le" :: "'a => 'a => 'b set => 'b set" ("(3INT _<=_./ _)" [0, 0, 10] 10)
- "_INTER_less" :: "'a => 'a => 'b set => 'b set" ("(3INT _<_./ _)" [0, 0, 10] 10)
-
-syntax (xsymbols)
- "_UNION_le" :: "'a => 'a => 'b set => 'b set" ("(3\<Union> _\<le>_./ _)" [0, 0, 10] 10)
- "_UNION_less" :: "'a => 'a => 'b set => 'b set" ("(3\<Union> _<_./ _)" [0, 0, 10] 10)
- "_INTER_le" :: "'a => 'a => 'b set => 'b set" ("(3\<Inter> _\<le>_./ _)" [0, 0, 10] 10)
- "_INTER_less" :: "'a => 'a => 'b set => 'b set" ("(3\<Inter> _<_./ _)" [0, 0, 10] 10)
-
-syntax (latex output)
- "_UNION_le" :: "'a \<Rightarrow> 'a => 'b set => 'b set" ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)
- "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set" ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)
- "_INTER_le" :: "'a \<Rightarrow> 'a => 'b set => 'b set" ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)
- "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set" ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 10)
-
-translations
- "UN i<=n. A" == "UN i:{..n}. A"
- "UN i<n. A" == "UN i:{..<n}. A"
- "INT i<=n. A" == "INT i:{..n}. A"
- "INT i<n. A" == "INT i:{..<n}. A"
-
-
-subsection {* Various equivalences *}
-
-lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
-by (simp add: lessThan_def)
-
-lemma Compl_lessThan [simp]:
- "!!k:: 'a::linorder. -lessThan k = atLeast k"
-apply (auto simp add: lessThan_def atLeast_def)
-done
-
-lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
-by auto
-
-lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
-by (simp add: greaterThan_def)
-
-lemma Compl_greaterThan [simp]:
- "!!k:: 'a::linorder. -greaterThan k = atMost k"
- by (auto simp add: greaterThan_def atMost_def)
-
-lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
-apply (subst Compl_greaterThan [symmetric])
-apply (rule double_complement)
-done
-
-lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
-by (simp add: atLeast_def)
-
-lemma Compl_atLeast [simp]:
- "!!k:: 'a::linorder. -atLeast k = lessThan k"
- by (auto simp add: lessThan_def atLeast_def)
-
-lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
-by (simp add: atMost_def)
-
-lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
-by (blast intro: order_antisym)
-
-
-subsection {* Logical Equivalences for Set Inclusion and Equality *}
-
-lemma atLeast_subset_iff [iff]:
- "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
-by (blast intro: order_trans)
-
-lemma atLeast_eq_iff [iff]:
- "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
-by (blast intro: order_antisym order_trans)
-
-lemma greaterThan_subset_iff [iff]:
- "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
-apply (auto simp add: greaterThan_def)
- apply (subst linorder_not_less [symmetric], blast)
-done
-
-lemma greaterThan_eq_iff [iff]:
- "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
-apply (rule iffI)
- apply (erule equalityE)
- apply simp_all
-done
-
-lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
-by (blast intro: order_trans)
-
-lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
-by (blast intro: order_antisym order_trans)
-
-lemma lessThan_subset_iff [iff]:
- "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
-apply (auto simp add: lessThan_def)
- apply (subst linorder_not_less [symmetric], blast)
-done
-
-lemma lessThan_eq_iff [iff]:
- "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
-apply (rule iffI)
- apply (erule equalityE)
- apply simp_all
-done
-
-lemma lessThan_strict_subset_iff:
- fixes m n :: "'a::linorder"
- shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
- by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
-
-subsection {*Two-sided intervals*}
-
-context ord
-begin
-
-lemma greaterThanLessThan_iff [simp,no_atp]:
- "(i : {l<..<u}) = (l < i & i < u)"
-by (simp add: greaterThanLessThan_def)
-
-lemma atLeastLessThan_iff [simp,no_atp]:
- "(i : {l..<u}) = (l <= i & i < u)"
-by (simp add: atLeastLessThan_def)
-
-lemma greaterThanAtMost_iff [simp,no_atp]:
- "(i : {l<..u}) = (l < i & i <= u)"
-by (simp add: greaterThanAtMost_def)
-
-lemma atLeastAtMost_iff [simp,no_atp]:
- "(i : {l..u}) = (l <= i & i <= u)"
-by (simp add: atLeastAtMost_def)
-
-text {* The above four lemmas could be declared as iffs. Unfortunately this
-breaks many proofs. Since it only helps blast, it is better to leave well
-alone *}
-
-end
-
-subsubsection{* Emptyness, singletons, subset *}
-
-context order
-begin
-
-lemma atLeastatMost_empty[simp]:
- "b < a \<Longrightarrow> {a..b} = {}"
-by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
-
-lemma atLeastatMost_empty_iff[simp]:
- "{a..b} = {} \<longleftrightarrow> (~ a <= b)"
-by auto (blast intro: order_trans)
-
-lemma atLeastatMost_empty_iff2[simp]:
- "{} = {a..b} \<longleftrightarrow> (~ a <= b)"
-by auto (blast intro: order_trans)
-
-lemma atLeastLessThan_empty[simp]:
- "b <= a \<Longrightarrow> {a..<b} = {}"
-by(auto simp: atLeastLessThan_def)
-
-lemma atLeastLessThan_empty_iff[simp]:
- "{a..<b} = {} \<longleftrightarrow> (~ a < b)"
-by auto (blast intro: le_less_trans)
-
-lemma atLeastLessThan_empty_iff2[simp]:
- "{} = {a..<b} \<longleftrightarrow> (~ a < b)"
-by auto (blast intro: le_less_trans)
-
-lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
-by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
-
-lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"
-by auto (blast intro: less_le_trans)
-
-lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"
-by auto (blast intro: less_le_trans)
-
-lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
-by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
-
-lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
-by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
-
-lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp
-
-lemma atLeastatMost_subset_iff[simp]:
- "{a..b} <= {c..d} \<longleftrightarrow> (~ a <= b) | c <= a & b <= d"
-unfolding atLeastAtMost_def atLeast_def atMost_def
-by (blast intro: order_trans)
-
-lemma atLeastatMost_psubset_iff:
- "{a..b} < {c..d} \<longleftrightarrow>
- ((~ a <= b) | c <= a & b <= d & (c < a | b < d)) & c <= d"
-by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)
-
-lemma atLeastAtMost_singleton_iff[simp]:
- "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"
-proof
- assume "{a..b} = {c}"
- hence "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
- moreover with `{a..b} = {c}` have "c \<le> a \<and> b \<le> c" by auto
- ultimately show "a = b \<and> b = c" by auto
-qed simp
-
-end
-
-context dense_linorder
-begin
-
-lemma greaterThanLessThan_empty_iff[simp]:
- "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
- using dense[of a b] by (cases "a < b") auto
-
-lemma greaterThanLessThan_empty_iff2[simp]:
- "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
- using dense[of a b] by (cases "a < b") auto
-
-lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
- "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
- using dense[of "max a d" "b"]
- by (force simp: subset_eq Ball_def not_less[symmetric])
-
-lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
- "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
- using dense[of "a" "min c b"]
- by (force simp: subset_eq Ball_def not_less[symmetric])
-
-lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
- "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
- using dense[of "a" "min c b"] dense[of "max a d" "b"]
- by (force simp: subset_eq Ball_def not_less[symmetric])
-
-lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
- "{a .. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"
- using dense[of "max a d" "b"]
- by (force simp: subset_eq Ball_def not_less[symmetric])
-
-lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
- "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
- using dense[of "a" "min c b"]
- by (force simp: subset_eq Ball_def not_less[symmetric])
-
-lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
- "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
- using dense[of "a" "min c b"] dense[of "max a d" "b"]
- by (force simp: subset_eq Ball_def not_less[symmetric])
-
-end
-
-lemma (in linorder) atLeastLessThan_subset_iff:
- "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
-apply (auto simp:subset_eq Ball_def)
-apply(frule_tac x=a in spec)
-apply(erule_tac x=d in allE)
-apply (simp add: less_imp_le)
-done
-
-lemma atLeastLessThan_inj:
- fixes a b c d :: "'a::linorder"
- assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
- shows "a = c" "b = d"
-using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
-
-lemma atLeastLessThan_eq_iff:
- fixes a b c d :: "'a::linorder"
- assumes "a < b" "c < d"
- shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
- using atLeastLessThan_inj assms by auto
-
-subsubsection {* Intersection *}
-
-context linorder
-begin
-
-lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
-by auto
-
-lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
-by auto
-
-lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
-by auto
-
-lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
-by auto
-
-lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
-by auto
-
-lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
-by auto
-
-lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
-by auto
-
-lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
-by auto
-
-end
-
-
-subsection {* Intervals of natural numbers *}
-
-subsubsection {* The Constant @{term lessThan} *}
-
-lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
-by (simp add: lessThan_def)
-
-lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
-by (simp add: lessThan_def less_Suc_eq, blast)
-
-text {* The following proof is convenient in induction proofs where
-new elements get indices at the beginning. So it is used to transform
-@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}
-
-lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
-proof safe
- fix x assume "x < Suc n" "x \<notin> Suc ` {..<n}"
- then have "x \<noteq> Suc (x - 1)" by auto
- with `x < Suc n` show "x = 0" by auto
-qed
-
-lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
-by (simp add: lessThan_def atMost_def less_Suc_eq_le)
-
-lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
-by blast
-
-subsubsection {* The Constant @{term greaterThan} *}
-
-lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
-apply (simp add: greaterThan_def)
-apply (blast dest: gr0_conv_Suc [THEN iffD1])
-done
-
-lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
-apply (simp add: greaterThan_def)
-apply (auto elim: linorder_neqE)
-done
-
-lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
-by blast
-
-subsubsection {* The Constant @{term atLeast} *}
-
-lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
-by (unfold atLeast_def UNIV_def, simp)
-
-lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
-apply (simp add: atLeast_def)
-apply (simp add: Suc_le_eq)
-apply (simp add: order_le_less, blast)
-done
-
-lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
- by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
-
-lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
-by blast
-
-subsubsection {* The Constant @{term atMost} *}
-
-lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
-by (simp add: atMost_def)
-
-lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
-apply (simp add: atMost_def)
-apply (simp add: less_Suc_eq order_le_less, blast)
-done
-
-lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
-by blast
-
-subsubsection {* The Constant @{term atLeastLessThan} *}
-
-text{*The orientation of the following 2 rules is tricky. The lhs is
-defined in terms of the rhs. Hence the chosen orientation makes sense
-in this theory --- the reverse orientation complicates proofs (eg
-nontermination). But outside, when the definition of the lhs is rarely
-used, the opposite orientation seems preferable because it reduces a
-specific concept to a more general one. *}
-
-lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
-by(simp add:lessThan_def atLeastLessThan_def)
-
-lemma atLeast0AtMost: "{0..n::nat} = {..n}"
-by(simp add:atMost_def atLeastAtMost_def)
-
-declare atLeast0LessThan[symmetric, code_unfold]
- atLeast0AtMost[symmetric, code_unfold]
-
-lemma atLeastLessThan0: "{m..<0::nat} = {}"
-by (simp add: atLeastLessThan_def)
-
-subsubsection {* Intervals of nats with @{term Suc} *}
-
-text{*Not a simprule because the RHS is too messy.*}
-lemma atLeastLessThanSuc:
- "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
-by (auto simp add: atLeastLessThan_def)
-
-lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
-by (auto simp add: atLeastLessThan_def)
-(*
-lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
-by (induct k, simp_all add: atLeastLessThanSuc)
-
-lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
-by (auto simp add: atLeastLessThan_def)
-*)
-lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
- by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
-
-lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
- by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
- greaterThanAtMost_def)
-
-lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
- by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
- greaterThanLessThan_def)
-
-lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
-by (auto simp add: atLeastAtMost_def)
-
-lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
-by auto
-
-text {* The analogous result is useful on @{typ int}: *}
-(* here, because we don't have an own int section *)
-lemma atLeastAtMostPlus1_int_conv:
- "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
- by (auto intro: set_eqI)
-
-lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
- apply (induct k)
- apply (simp_all add: atLeastLessThanSuc)
- done
-
-subsubsection {* Image *}
-
-lemma image_add_atLeastAtMost:
- "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
-proof
- show "?A \<subseteq> ?B" by auto
-next
- show "?B \<subseteq> ?A"
- proof
- fix n assume a: "n : ?B"
- hence "n - k : {i..j}" by auto
- moreover have "n = (n - k) + k" using a by auto
- ultimately show "n : ?A" by blast
- qed
-qed
-
-lemma image_add_atLeastLessThan:
- "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
-proof
- show "?A \<subseteq> ?B" by auto
-next
- show "?B \<subseteq> ?A"
- proof
- fix n assume a: "n : ?B"
- hence "n - k : {i..<j}" by auto
- moreover have "n = (n - k) + k" using a by auto
- ultimately show "n : ?A" by blast
- qed
-qed
-
-corollary image_Suc_atLeastAtMost[simp]:
- "Suc ` {i..j} = {Suc i..Suc j}"
-using image_add_atLeastAtMost[where k="Suc 0"] by simp
-
-corollary image_Suc_atLeastLessThan[simp]:
- "Suc ` {i..<j} = {Suc i..<Suc j}"
-using image_add_atLeastLessThan[where k="Suc 0"] by simp
-
-lemma image_add_int_atLeastLessThan:
- "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
- apply (auto simp add: image_def)
- apply (rule_tac x = "x - l" in bexI)
- apply auto
- done
-
-lemma image_minus_const_atLeastLessThan_nat:
- fixes c :: nat
- shows "(\<lambda>i. i - c) ` {x ..< y} =
- (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
- (is "_ = ?right")
-proof safe
- fix a assume a: "a \<in> ?right"
- show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
- proof cases
- assume "c < y" with a show ?thesis
- by (auto intro!: image_eqI[of _ _ "a + c"])
- next
- assume "\<not> c < y" with a show ?thesis
- by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
- qed
-qed auto
-
-context ordered_ab_group_add
-begin
-
-lemma
- fixes x :: 'a
- shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
- and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
-proof safe
- fix y assume "y < -x"
- hence *: "x < -y" using neg_less_iff_less[of "-y" x] by simp
- have "- (-y) \<in> uminus ` {x<..}"
- by (rule imageI) (simp add: *)
- thus "y \<in> uminus ` {x<..}" by simp
-next
- fix y assume "y \<le> -x"
- have "- (-y) \<in> uminus ` {x..}"
- by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)
- thus "y \<in> uminus ` {x..}" by simp
-qed simp_all
-
-lemma
- fixes x :: 'a
- shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
- and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
-proof -
- have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
- and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
- thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
- by (simp_all add: image_image
- del: image_uminus_greaterThan image_uminus_atLeast)
-qed
-
-lemma
- fixes x :: 'a
- shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
- and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
- and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
- and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
- by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
- greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
-end
-
-subsubsection {* Finiteness *}
-
-lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
- by (induct k) (simp_all add: lessThan_Suc)
-
-lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
- by (induct k) (simp_all add: atMost_Suc)
-
-lemma finite_greaterThanLessThan [iff]:
- fixes l :: nat shows "finite {l<..<u}"
-by (simp add: greaterThanLessThan_def)
-
-lemma finite_atLeastLessThan [iff]:
- fixes l :: nat shows "finite {l..<u}"
-by (simp add: atLeastLessThan_def)
-
-lemma finite_greaterThanAtMost [iff]:
- fixes l :: nat shows "finite {l<..u}"
-by (simp add: greaterThanAtMost_def)
-
-lemma finite_atLeastAtMost [iff]:
- fixes l :: nat shows "finite {l..u}"
-by (simp add: atLeastAtMost_def)
-
-text {* A bounded set of natural numbers is finite. *}
-lemma bounded_nat_set_is_finite:
- "(ALL i:N. i < (n::nat)) ==> finite N"
-apply (rule finite_subset)
- apply (rule_tac [2] finite_lessThan, auto)
-done
-
-text {* A set of natural numbers is finite iff it is bounded. *}
-lemma finite_nat_set_iff_bounded:
- "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
-proof
- assume f:?F show ?B
- using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
-next
- assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
-qed
-
-lemma finite_nat_set_iff_bounded_le:
- "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
-apply(simp add:finite_nat_set_iff_bounded)
-apply(blast dest:less_imp_le_nat le_imp_less_Suc)
-done
-
-lemma finite_less_ub:
- "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
-by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
-
-text{* Any subset of an interval of natural numbers the size of the
-subset is exactly that interval. *}
-
-lemma subset_card_intvl_is_intvl:
- "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
-proof cases
- assume "finite A"
- thus "PROP ?P"
- proof(induct A rule:finite_linorder_max_induct)
- case empty thus ?case by auto
- next
- case (insert b A)
- moreover hence "b ~: A" by auto
- moreover have "A <= {k..<k+card A}" and "b = k+card A"
- using `b ~: A` insert by fastforce+
- ultimately show ?case by auto
- qed
-next
- assume "~finite A" thus "PROP ?P" by simp
-qed
-
-
-subsubsection {* Proving Inclusions and Equalities between Unions *}
-
-lemma UN_le_eq_Un0:
- "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
-proof
- show "?A <= ?B"
- proof
- fix x assume "x : ?A"
- then obtain i where i: "i\<le>n" "x : M i" by auto
- show "x : ?B"
- proof(cases i)
- case 0 with i show ?thesis by simp
- next
- case (Suc j) with i show ?thesis by auto
- qed
- qed
-next
- show "?B <= ?A" by auto
-qed
-
-lemma UN_le_add_shift:
- "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
-proof
- show "?A <= ?B" by fastforce
-next
- show "?B <= ?A"
- proof
- fix x assume "x : ?B"
- then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
- hence "i-k\<le>n & x : M((i-k)+k)" by auto
- thus "x : ?A" by blast
- qed
-qed
-
-lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
- by (auto simp add: atLeast0LessThan)
-
-lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
- by (subst UN_UN_finite_eq [symmetric]) blast
-
-lemma UN_finite2_subset:
- "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) \<subseteq> (\<Union>n. B n)"
- apply (rule UN_finite_subset)
- apply (subst UN_UN_finite_eq [symmetric, of B])
- apply blast
- done
-
-lemma UN_finite2_eq:
- "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) = (\<Union>n. B n)"
- apply (rule subset_antisym)
- apply (rule UN_finite2_subset, blast)
- apply (rule UN_finite2_subset [where k=k])
- apply (force simp add: atLeastLessThan_add_Un [of 0])
- done
-
-
-subsubsection {* Cardinality *}
-
-lemma card_lessThan [simp]: "card {..<u} = u"
- by (induct u, simp_all add: lessThan_Suc)
-
-lemma card_atMost [simp]: "card {..u} = Suc u"
- by (simp add: lessThan_Suc_atMost [THEN sym])
-
-lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
- apply (subgoal_tac "card {l..<u} = card {..<u-l}")
- apply (erule ssubst, rule card_lessThan)
- apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
- apply (erule subst)
- apply (rule card_image)
- apply (simp add: inj_on_def)
- apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
- apply (rule_tac x = "x - l" in exI)
- apply arith
- done
-
-lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
- by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
-
-lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
- by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
-
-lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
- by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
-
-lemma ex_bij_betw_nat_finite:
- "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
-apply(drule finite_imp_nat_seg_image_inj_on)
-apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
-done
-
-lemma ex_bij_betw_finite_nat:
- "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
-by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
-
-lemma finite_same_card_bij:
- "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
-apply(drule ex_bij_betw_finite_nat)
-apply(drule ex_bij_betw_nat_finite)
-apply(auto intro!:bij_betw_trans)
-done
-
-lemma ex_bij_betw_nat_finite_1:
- "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
-by (rule finite_same_card_bij) auto
-
-lemma bij_betw_iff_card:
- assumes FIN: "finite A" and FIN': "finite B"
- shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
-using assms
-proof(auto simp add: bij_betw_same_card)
- assume *: "card A = card B"
- obtain f where "bij_betw f A {0 ..< card A}"
- using FIN ex_bij_betw_finite_nat by blast
- moreover obtain g where "bij_betw g {0 ..< card B} B"
- using FIN' ex_bij_betw_nat_finite by blast
- ultimately have "bij_betw (g o f) A B"
- using * by (auto simp add: bij_betw_trans)
- thus "(\<exists>f. bij_betw f A B)" by blast
-qed
-
-lemma inj_on_iff_card_le:
- assumes FIN: "finite A" and FIN': "finite B"
- shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
-proof (safe intro!: card_inj_on_le)
- assume *: "card A \<le> card B"
- obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
- using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
- moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
- using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
- ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
- hence "inj_on (g o f) A" using 1 comp_inj_on by blast
- moreover
- {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
- with 2 have "f ` A \<le> {0 ..< card B}" by blast
- hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
- }
- ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
-qed (insert assms, auto)
-
-subsection {* Intervals of integers *}
-
-lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
- by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
-
-lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
- by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
-
-lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
- "{l+1..<u} = {l<..<u::int}"
- by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
-
-subsubsection {* Finiteness *}
-
-lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
- {(0::int)..<u} = int ` {..<nat u}"
- apply (unfold image_def lessThan_def)
- apply auto
- apply (rule_tac x = "nat x" in exI)
- apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
- done
-
-lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
- apply (case_tac "0 \<le> u")
- apply (subst image_atLeastZeroLessThan_int, assumption)
- apply (rule finite_imageI)
- apply auto
- done
-
-lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
- apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
- apply (erule subst)
- apply (rule finite_imageI)
- apply (rule finite_atLeastZeroLessThan_int)
- apply (rule image_add_int_atLeastLessThan)
- done
-
-lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
- by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
-
-lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
- by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
-
-lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
- by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
-
-
-subsubsection {* Cardinality *}
-
-lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
- apply (case_tac "0 \<le> u")
- apply (subst image_atLeastZeroLessThan_int, assumption)
- apply (subst card_image)
- apply (auto simp add: inj_on_def)
- done
-
-lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
- apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
- apply (erule ssubst, rule card_atLeastZeroLessThan_int)
- apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
- apply (erule subst)
- apply (rule card_image)
- apply (simp add: inj_on_def)
- apply (rule image_add_int_atLeastLessThan)
- done
-
-lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
-apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
-apply (auto simp add: algebra_simps)
-done
-
-lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
-by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
-
-lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
-by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
-
-lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
-proof -
- have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
- with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
-qed
-
-lemma card_less:
-assumes zero_in_M: "0 \<in> M"
-shows "card {k \<in> M. k < Suc i} \<noteq> 0"
-proof -
- from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
- with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
-qed
-
-lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
-apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
-apply simp
-apply fastforce
-apply auto
-apply (rule inj_on_diff_nat)
-apply auto
-apply (case_tac x)
-apply auto
-apply (case_tac xa)
-apply auto
-apply (case_tac xa)
-apply auto
-done
-
-lemma card_less_Suc:
- assumes zero_in_M: "0 \<in> M"
- shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
-proof -
- from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
- hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
- by (auto simp only: insert_Diff)
- have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}" by auto
- from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
- apply (subst card_insert)
- apply simp_all
- apply (subst b)
- apply (subst card_less_Suc2[symmetric])
- apply simp_all
- done
- with c show ?thesis by simp
-qed
-
-
-subsection {*Lemmas useful with the summation operator setsum*}
-
-text {* For examples, see Algebra/poly/UnivPoly2.thy *}
-
-subsubsection {* Disjoint Unions *}
-
-text {* Singletons and open intervals *}
-
-lemma ivl_disj_un_singleton:
- "{l::'a::linorder} Un {l<..} = {l..}"
- "{..<u} Un {u::'a::linorder} = {..u}"
- "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
- "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
- "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
- "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
-by auto
-
-text {* One- and two-sided intervals *}
-
-lemma ivl_disj_un_one:
- "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
- "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
- "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
- "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
- "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
- "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
- "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
- "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
-by auto
-
-text {* Two- and two-sided intervals *}
-
-lemma ivl_disj_un_two:
- "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
- "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
- "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
- "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
- "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
- "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
- "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
- "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
-by auto
-
-lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
-
-subsubsection {* Disjoint Intersections *}
-
-text {* One- and two-sided intervals *}
-
-lemma ivl_disj_int_one:
- "{..l::'a::order} Int {l<..<u} = {}"
- "{..<l} Int {l..<u} = {}"
- "{..l} Int {l<..u} = {}"
- "{..<l} Int {l..u} = {}"
- "{l<..u} Int {u<..} = {}"
- "{l<..<u} Int {u..} = {}"
- "{l..u} Int {u<..} = {}"
- "{l..<u} Int {u..} = {}"
- by auto
-
-text {* Two- and two-sided intervals *}
-
-lemma ivl_disj_int_two:
- "{l::'a::order<..<m} Int {m..<u} = {}"
- "{l<..m} Int {m<..<u} = {}"
- "{l..<m} Int {m..<u} = {}"
- "{l..m} Int {m<..<u} = {}"
- "{l<..<m} Int {m..u} = {}"
- "{l<..m} Int {m<..u} = {}"
- "{l..<m} Int {m..u} = {}"
- "{l..m} Int {m<..u} = {}"
- by auto
-
-lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
-
-subsubsection {* Some Differences *}
-
-lemma ivl_diff[simp]:
- "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
-by(auto)
-
-
-subsubsection {* Some Subset Conditions *}
-
-lemma ivl_subset [simp,no_atp]:
- "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
-apply(auto simp:linorder_not_le)
-apply(rule ccontr)
-apply(insert linorder_le_less_linear[of i n])
-apply(clarsimp simp:linorder_not_le)
-apply(fastforce)
-done
-
-
-subsection {* Summation indexed over intervals *}
-
-syntax
- "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
- "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
- "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
- "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
-syntax (xsymbols)
- "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
- "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
- "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
- "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
-syntax (HTML output)
- "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
- "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
- "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
- "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
-syntax (latex_sum output)
- "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
- ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
- "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
- ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
- "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
- ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
- "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
- ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
-
-translations
- "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
- "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
- "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
- "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
-
-text{* The above introduces some pretty alternative syntaxes for
-summation over intervals:
-\begin{center}
-\begin{tabular}{lll}
-Old & New & \LaTeX\\
-@{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
-@{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
-@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
-@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
-\end{tabular}
-\end{center}
-The left column shows the term before introduction of the new syntax,
-the middle column shows the new (default) syntax, and the right column
-shows a special syntax. The latter is only meaningful for latex output
-and has to be activated explicitly by setting the print mode to
-@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
-antiquotations). It is not the default \LaTeX\ output because it only
-works well with italic-style formulae, not tt-style.
-
-Note that for uniformity on @{typ nat} it is better to use
-@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
-not provide all lemmas available for @{term"{m..<n}"} also in the
-special form for @{term"{..<n}"}. *}
-
-text{* This congruence rule should be used for sums over intervals as
-the standard theorem @{text[source]setsum_cong} does not work well
-with the simplifier who adds the unsimplified premise @{term"x:B"} to
-the context. *}
-
-lemma setsum_ivl_cong:
- "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
- setsum f {a..<b} = setsum g {c..<d}"
-by(rule setsum_cong, simp_all)
-
-(* FIXME why are the following simp rules but the corresponding eqns
-on intervals are not? *)
-
-lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
-by (simp add:atMost_Suc add_ac)
-
-lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
-by (simp add:lessThan_Suc add_ac)
-
-lemma setsum_cl_ivl_Suc[simp]:
- "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
-by (auto simp:add_ac atLeastAtMostSuc_conv)
-
-lemma setsum_op_ivl_Suc[simp]:
- "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
-by (auto simp:add_ac atLeastLessThanSuc)
-(*
-lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
- (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
-by (auto simp:add_ac atLeastAtMostSuc_conv)
-*)
-
-lemma setsum_head:
- fixes n :: nat
- assumes mn: "m <= n"
- shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
-proof -
- from mn
- have "{m..n} = {m} \<union> {m<..n}"
- by (auto intro: ivl_disj_un_singleton)
- hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
- by (simp add: atLeast0LessThan)
- also have "\<dots> = ?rhs" by simp
- finally show ?thesis .
-qed
-
-lemma setsum_head_Suc:
- "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
-by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
-
-lemma setsum_head_upt_Suc:
- "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
-apply(insert setsum_head_Suc[of m "n - Suc 0" f])
-apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
-done
-
-lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
- shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
-proof-
- have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
- thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
- atLeastSucAtMost_greaterThanAtMost)
-qed
-
-lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
- setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
-by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
-
-lemma setsum_diff_nat_ivl:
-fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
-shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
- setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
-using setsum_add_nat_ivl [of m n p f,symmetric]
-apply (simp add: add_ac)
-done
-
-lemma setsum_natinterval_difff:
- fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
- shows "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
- (if m <= n then f m - f(n + 1) else 0)"
-by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
-
-lemma setsum_restrict_set':
- "finite A \<Longrightarrow> setsum f {x \<in> A. x \<in> B} = (\<Sum>x\<in>A. if x \<in> B then f x else 0)"
- by (simp add: setsum_restrict_set [symmetric] Int_def)
-
-lemma setsum_restrict_set'':
- "finite A \<Longrightarrow> setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x then f x else 0)"
- by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])
-
-lemma setsum_setsum_restrict:
- "finite S \<Longrightarrow> finite T \<Longrightarrow>
- setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y \<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
- by (simp add: setsum_restrict_set'') (rule setsum_commute)
-
-lemma setsum_image_gen: assumes fS: "finite S"
- shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
-proof-
- { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
- hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
- by simp
- also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
- by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
- finally show ?thesis .
-qed
-
-lemma setsum_le_included:
- fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
- assumes "finite s" "finite t"
- and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
- shows "setsum f s \<le> setsum g t"
-proof -
- have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
- proof (rule setsum_mono)
- fix y assume "y \<in> s"
- with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
- with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
- using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
- by (auto intro!: setsum_mono2)
- qed
- also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
- using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
- also have "... \<le> setsum g t"
- using assms by (auto simp: setsum_image_gen[symmetric])
- finally show ?thesis .
-qed
-
-lemma setsum_multicount_gen:
- assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
- shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
-proof-
- have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
- also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
- using assms(3) by auto
- finally show ?thesis .
-qed
-
-lemma setsum_multicount:
- assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
- shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
-proof-
- have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
- also have "\<dots> = ?r" by(simp add: mult_commute)
- finally show ?thesis by auto
-qed
-
-
-subsection{* Shifting bounds *}
-
-lemma setsum_shift_bounds_nat_ivl:
- "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
-by (induct "n", auto simp:atLeastLessThanSuc)
-
-lemma setsum_shift_bounds_cl_nat_ivl:
- "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
-apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
-apply (simp add:image_add_atLeastAtMost o_def)
-done
-
-corollary setsum_shift_bounds_cl_Suc_ivl:
- "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
-by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
-
-corollary setsum_shift_bounds_Suc_ivl:
- "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
-by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
-
-lemma setsum_shift_lb_Suc0_0:
- "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
-by(simp add:setsum_head_Suc)
-
-lemma setsum_shift_lb_Suc0_0_upt:
- "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
-apply(cases k)apply simp
-apply(simp add:setsum_head_upt_Suc)
-done
-
-subsection {* The formula for geometric sums *}
-
-lemma geometric_sum:
- assumes "x \<noteq> 1"
- shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
-proof -
- from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
- moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
- proof (induct n)
- case 0 then show ?case by simp
- next
- case (Suc n)
- moreover with `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp
- ultimately show ?case by (simp add: field_simps divide_inverse)
- qed
- ultimately show ?thesis by simp
-qed
-
-
-subsection {* The formula for arithmetic sums *}
-
-lemma gauss_sum:
- "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) =
- of_nat n*((of_nat n)+1)"
-proof (induct n)
- case 0
- show ?case by simp
-next
- case (Suc n)
- then show ?case
- by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)
- (* FIXME: make numeral cancellation simprocs work for semirings *)
-qed
-
-theorem arith_series_general:
- "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
- of_nat n * (a + (a + of_nat(n - 1)*d))"
-proof cases
- assume ngt1: "n > 1"
- let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
- have
- "(\<Sum>i\<in>{..<n}. a+?I i*d) =
- ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
- by (rule setsum_addf)
- also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
- also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
- unfolding One_nat_def
- by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
- also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
- by (simp add: algebra_simps)
- also from ngt1 have "{1..<n} = {1..n - 1}"
- by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
- also from ngt1
- have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"
- by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
- (simp add: mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
- finally show ?thesis
- unfolding mult_2 by (simp add: algebra_simps)
-next
- assume "\<not>(n > 1)"
- hence "n = 1 \<or> n = 0" by auto
- thus ?thesis by (auto simp: mult_2)
-qed
-
-lemma arith_series_nat:
- "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
-proof -
- have
- "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
- of_nat(n) * (a + (a + of_nat(n - 1)*d))"
- by (rule arith_series_general)
- thus ?thesis
- unfolding One_nat_def by auto
-qed
-
-lemma arith_series_int:
- "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"
- by (fact arith_series_general) (* FIXME: duplicate *)
-
-lemma sum_diff_distrib:
- fixes P::"nat\<Rightarrow>nat"
- shows
- "\<forall>x. Q x \<le> P x \<Longrightarrow>
- (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
-proof (induct n)
- case 0 show ?case by simp
-next
- case (Suc n)
-
- let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
- let ?rhs = "\<Sum>x<n. P x - Q x"
-
- from Suc have "?lhs = ?rhs" by simp
- moreover
- from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
- moreover
- from Suc have
- "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
- by (subst diff_diff_left[symmetric],
- subst diff_add_assoc2)
- (auto simp: diff_add_assoc2 intro: setsum_mono)
- ultimately
- show ?case by simp
-qed
-
-subsection {* Products indexed over intervals *}
-
-syntax
- "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
- "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
- "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
- "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
-syntax (xsymbols)
- "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
- "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
- "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
- "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
-syntax (HTML output)
- "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
- "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
- "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
- "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
-syntax (latex_prod output)
- "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
- ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
- "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
- ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
- "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
- ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
- "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
- ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
-
-translations
- "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
- "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
- "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
- "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
-
-subsection {* Transfer setup *}
-
-lemma transfer_nat_int_set_functions:
- "{..n} = nat ` {0..int n}"
- "{m..n} = nat ` {int m..int n}" (* need all variants of these! *)
- apply (auto simp add: image_def)
- apply (rule_tac x = "int x" in bexI)
- apply auto
- apply (rule_tac x = "int x" in bexI)
- apply auto
- done
-
-lemma transfer_nat_int_set_function_closures:
- "x >= 0 \<Longrightarrow> nat_set {x..y}"
- by (simp add: nat_set_def)
-
-declare transfer_morphism_nat_int[transfer add
- return: transfer_nat_int_set_functions
- transfer_nat_int_set_function_closures
-]
-
-lemma transfer_int_nat_set_functions:
- "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
- by (simp only: is_nat_def transfer_nat_int_set_functions
- transfer_nat_int_set_function_closures
- transfer_nat_int_set_return_embed nat_0_le
- cong: transfer_nat_int_set_cong)
-
-lemma transfer_int_nat_set_function_closures:
- "is_nat x \<Longrightarrow> nat_set {x..y}"
- by (simp only: transfer_nat_int_set_function_closures is_nat_def)
-
-declare transfer_morphism_int_nat[transfer add
- return: transfer_int_nat_set_functions
- transfer_int_nat_set_function_closures
-]
-
-end